No comma left unlabelled, no temperament left undescribed
If you’ve ever stumbled onto the Xenharmonic Wiki — the sprawling, community-built encyclopedia of alternative tuning systems — you may have noticed something odd. A huge number of its entries carry a small note: “name proposed by Xenllium.” Temperaments, commas, scales: hundreds of them, named by a single person. Who is this person, and why does any of it matter?
This article is an introduction to the work of Xenllium, a Japanese microtonal music theorist who joined the Xenharmonic Wiki in November 2018 and has since proposed names for well over 500 theoretical objects in microtonal music. The name “Xenllium” is itself a portmanteau of “Xenharmonic” and “Trillium” — a flower, and a fitting emblem for someone who finds beauty in mathematical structures most musicians never know exist.
First: What Is Microtonal Music, Anyway?
Before we can appreciate what Xenllium does, you need a quick map of the territory.
Western music, as you probably know it, uses 12 equal steps per octave — this is called 12edo (12 equal divisions of the octave), or 12-tone equal temperament. Every piano key, every fret on a standard guitar, every note in your DAW corresponds to one of those 12 equal steps.
But 12 is just one choice. You could divide the octave into 19 equal steps (19edo), or 31 (31edo), or 53 (53edo), or 171. Each produces a different universe of intervals, chords, and scales. Some of these — particularly 19, 31, and 53 — have been seriously explored by composers and theorists for centuries. Others are more exotic.
Beyond equal temperaments entirely, there’s just intonation: tuning notes to exact whole-number frequency ratios (like 3/2 for a perfect fifth, or 5/4 for a pure major third). When you stack just intervals in different ways, you get an infinite variety of possible tuning systems, each with its own internal logic. The study and practice of all these alternatives is called xenharmonic music.
The Building Blocks: Temperaments, Commas, and Scales
Three of the main objects Xenllium names are temperaments, commas, and scales. Here’s what those are.
Commas are tiny intervals that arise from the mismatch between different just intervals. The most famous example is the Pythagorean comma: if you stack 12 pure perfect fifths (ratio 3/2), you end up slightly higher than 7 octaves. The gap — about 23.5 cents, or roughly a quarter of a semitone — is the Pythagorean comma. Another important one is the syntonic comma (about 21.5 cents), which is the difference between four pure fifths and a pure major third.
Commas matter because they define temperaments. A regular temperament is a tuning system that deliberately “tempers out” one or more commas — that is, it treats the gap as zero and maps those two intervals to the same pitch. Meantone temperament, which underlies most Western music from the Renaissance through the 19th century, tempers out the syntonic comma. This makes all whole tones the same size (that’s where “mean” comes from), and it gives you pure or near-pure major thirds.
Different choices of which commas to temper out lead to completely different temperaments, with different harmonic personalities. Magic temperament is generated by a major third rather than a fifth. Miracle temperament can stack up pure approximations of the 7th harmonic. There are thousands of mathematically distinct temperaments, each with its own structure.
MOS scales (Moment of Symmetry scales) are the natural scales that arise within a given temperament — analogous to how the familiar major scale arises naturally within meantone. A MOS scale consists of two step sizes, arranged in a pattern that has exactly as many positions as it has notes.
What Xenllium Does
The Xenharmonic Wiki documents all of this. It catalogs EDOs, temperaments, commas, and scales, and gives them names that theorists can use to talk about them. Names matter: it’s much easier to discuss “magic temperament” than to say “the rank-2 temperament that tempers out the comma 3125/3072 and has a major third as its generator.”
Xenllium’s contribution is naming. A lot of naming. The page attached to this article is a list of everything Xenllium has named since 2019, and it is staggering in its scope.
Here’s a sample from just one month — May 2021:
- Temperaments: Quadrant, Quintilipyth, Helenoid, Ponta, Pontic, Bipont, Counterbipont, Quadrapont, Decimetra, Stockhausenic, Decavish, Bronzoid, Ennealimmia, Paralytic, Cotoneum, Fog, Countermiracle, Paraklein, Countermiraculous, Counterbenediction, Counterrevelation, Septant, Terminal, Semiterm, Sedia, Marthirds, Counterkleismic, Counterlytic, Geb, Hemidimfi, Maquiloid, Paramity
That’s in a single month, in a single year, and it’s not even close to his busiest period. By 2026, Xenllium has proposed names for well over 400 temperaments, more than 200 commas, and over two dozen scales.
The Names Themselves: A Pattern
One of the delightful things about studying Xenllium’s naming is the systematic creativity on display. The names follow recognizable patterns:
Names that indicate mathematical structure: “Hemiquindromeda,” “Undetrita,” “Semihemiwür,” “Biquartonic.” The prefixes hemi-, semi-, bi-, tri-, quad-, quint- indicate halving, quartering, or multiplying the generator count. “Hem” or “hemi” often means half; “un-” or “und-” can signal 11 (from Latin undecim).
Names borrowed from mythology, nature, and culture: “Cobalt,” “Rubidium,” “Terbium,” “Goldmine,” “Mineral,” “Ore” — a cluster of mineralogy-themed names proposed in 2021–22. “Narayana,” “Turkey,” “Eagle,” “Beetle,” “Witcher.” “Raccoon” (May 2024). “Magikarp” (April 2025), which targets the prime 43 in the 2.3.43 subgroup — a completely alien mathematical corner of tuning theory, named after a Pokémon.
Names that reference other temperaments with a modifier: “Countermiracle,” “Counterkleismic,” “Counterbipont” — these are the “mirror image” or dual version of existing temperaments, generated by the complement of the original generator.
Names from the Japanese document included in the PDF: Xenllium has written documents in Japanese about equal temperament theory, and his comma names sometimes reflect Japanese aesthetics — “Naiad,” “Seesaw,” “Heavy windmill,” “Light windmill” all appearing in early 2026.
The Commas: A Taxonomy of Tiny Gaps
Xenllium’s comma names are perhaps the most impressive part of the project. Here is a small selection, with their ratios:
The hewuermera (589824/588245) — a ratio involving primes 2, 3, 5, and 7 that sits just under 5 cents. The name follows a convention used in microtonal theory called color notation, where the prime components of a ratio are translated into color-based syllables.
The chromatisma (640000000000000000 / 635585924776181463) — an astronomically complex ratio involving many stacked primes, yet it has a clean mathematical meaning in the context of tuning theory.
The triwellisma (235298/234375), the quadrawellisma (67528125/67108864) — a family of commas related to “wellisma,” suggesting relationships to well-temperament theory.
More recently, Xenllium has been naming commas in the undevicesimal (19-limit) and trigesimal tertial (43-limit) ranges — the frontiers of just intonation theory, involving prime harmonics that most musicians have never thought about. A “43-limit” interval involves the 43rd harmonic of the overtone series, which sits between an ordinary minor third and a major third, wildly out of tune relative to anything on a standard keyboard.
In 2025, Xenllium named a whole collection of commas in the 47-limit and 53-limit: the tricema (93/92), the scyllisma(155/154), the eudorisma (217/216), the californisma (341/340), the ululisma (714/713). These are all superparticular ratios — ratios of the form (n+1)/n — which are especially elegant mathematically, because they represent the smallest possible interval between two consecutive harmonics at a given level.
The Xentwelve Tuning: Where Theory Meets Music
Xenllium is not only a theorist. He has composed music and designed tuning systems, and the most developed of these is his Xentwelve tuning.
Xentwelve is a 12-note circulating scale — meaning it covers all 12 pitch classes of a standard keyboard, and you can play in any key without running into a “wolf fifth” that sounds terrible. But unlike 12edo, the intervals are not equal. Xentwelve is close to 1/3-comma meantone in the natural keys (giving pure or near-pure major thirds in C and G major) and approaches Pythagorean tuning in the remote keys.
Specifically, the scale uses three different sizes of fifth:
- Eight pure fifths (ratio exactly 3/2) on the common white-key pairs like C–G, E–B, F–C
- Three 1/3-comma meantone fifths (slightly narrower) on D–A, G–D, and A–E
- One schisma-compressed fifth (ratio 16384/10935, very close to 700 cents) on G♯–D♯/A♭–E♭
The result is a scale with two pure major thirds (C–E and G–B) and one pure minor third (E–G), giving a perfectly pure C major triad and a perfectly pure E minor triad. This is the kind of thing that a Renaissance organist would have loved, achieved here through careful mathematical design.
You can hear demonstrations of Xentwelve on Xenllium’s YouTube channel (カタギリノエンレイソウ), along with his xenharmonic composition “Magical life,” written in a 19-tone pure-fifth magic scale.
The Notation System
Xenllium has also designed an original microtonal notation system, described on the Xenharmonic Wiki. The system is valid for any tuning where the chromatic semitone spans 1 to 16 steps — covering a huge range of EDOs from 12 out to roughly 200. It builds on familiar foundations:
- Standard sharp/flat notation for semitone alterations
- Stein-Zimmermann quartertone accidentals (the “semisharp” and “semiflat” symbols) for half-semitone alterations
- Symbols borrowed from Helmholtz-Ellis notation for finer subdivisions: quarter-semitone, sixth-semitone, eighth-semitone, down to sixteenth-semitone
The system is designed to be learnable and systematic rather than requiring entirely new symbols for each new EDO — a practical philosophy that reflects Xenllium’s broader approach of building on existing structures rather than starting from scratch.
31-Tone Circulating Scales
Alongside Xentwelve, Xenllium designed two 31-note circulating scales called Xenthirtyone I and Xenthirtyone II, based on 31edo. Where Xentwelve stays close to standard keyboard practice, Xenthirtyone ventures into genuinely xenharmonic territory.
31edo is a remarkable tuning system. Its major third (about 387 cents) is nearly pure — much better than 12edo’s 400 cents, which is 14 cents sharp of the pure 5/4 ratio. Its minor seventh approximates the 7th harmonic of the overtone series much better than 12edo does, giving it a characteristic “barbershop” quality that 12edo can’t reproduce. The Xenthirtyone scales use three sizes of major third, with most thirds pure and a few stretched, to maintain internal consistency across all 31 keys.
Why Does This Matter?
You might wonder: why does any of this need names? Who is actually using terms like “Scyllisma” or “Moctpine” in daily musical practice?
The answer is: the microtonal theory community, which uses these names actively on the Xenharmonic Wiki, in Discord servers, in tuning software like Scale Workshop, and in the design of new instruments and compositions. When a composer wants to write music in a particular temperament, knowing its name allows them to quickly look up what EDOs support it, what commas it tempers out, what its characteristic intervals are, and what other composers have done with it.
Names are how communities accumulate knowledge. Before Xenllium’s contributions, many of the mathematical objects they named existed in a theoretical limbo — discoverable by calculation but not discussable by reference. Xenllium has, over seven years of sustained effort, converted a large part of that limbo into a named, searchable, citable landscape.
There’s also something more personal here. The very act of naming — especially naming things like “Raccoon,” “Magikarp,” “Heavy windmill,” and “The sadness” (the title of one of his compositions) — reflects someone who finds genuine delight in this material. Microtonal theory can easily become arid and purely mathematical. Xenllium’s naming sensibility keeps it human.
Where to Start If You’re Curious
If this has piqued your interest, here are some entry points:
- The Xenharmonic Wiki main page is the central hub. Start with the articles on regular temperament and EDO for foundational concepts.
- Scale Workshop is a free browser-based tool that lets you design and hear microtonal scales immediately, no special hardware needed.
- For Xenllium’s work specifically, search the Xenharmonic Wiki for any of the temperament or comma names listed in this article — most have their own dedicated pages.
- Xenllium’s YouTube channel (カタギリノエンレイソウ) has recordings in Xentwelve tuning and magic temperament, which gives you an immediate sense of what this theory sounds like in practice.
The world of microtonal music is vast, ancient, mathematically deep, and full of people doing genuinely original work in their spare time. Xenllium is one of the most prolific contributors to its theoretical vocabulary. And somewhere out there, in a corner of the xenharmonic universe named “Raccoon” or “Magikarp,” there is music waiting to be written.
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